Question

How do you find the end behavior and state the possible number of x intercepts and the value of the y intercept given `y=x^3-4x`?

Answer 1

See the explanation below.

Explanation:

First, graph the function `f(x)=x^3-4x`.
graph{x^3-4x [-10, 10, -5, 5]}

From the graph, you can see as `x -> oo`, `f(x) -> oo`.
As `x->-oo`, `f(x) -> -oo`.

You can also write this using limits:

`lim_(x->oo) f(x) = oo`

`lim_(x->-oo) f(x) = -oo`

There are 3 `x`-intercepts here because there are 3 points where the graph intercepts the `x`-axis.

Similarly, you can find the `y`-intercept graphically if you find the point where the graph intercepts the `y`-axis. In this case, the `y`-intercept is 0.

If you want to find the `y`-intercept algebraically, substitute 0 in for x.

`y=x^3-4x`
`y=0^3-4(0)`
`y=0`